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\author{学号 \underline{\hspace{4cm}} \hspace{1cm} 姓名 \underline{\hspace{4cm}} }
\title{实变函数练习5.5-5.6 \\ 勒贝格积分与黎曼积分、勒贝格积分的几何意义}
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\date{2024 年 5 月 27 日}
%\date{March 9, 2021}

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\begin{document}

\maketitle

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\begin{enumerate}

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%\item  %Problem 01
%解释勒贝格积分是黎曼积分的推广，但不是黎曼反常积分的推广。
%
%\vspace{0.1cm}

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\item  %Problem 02、S5.5定理1
设函数 $f(x)$ 是区间 $[a,b]$ 上的有界函数，则 $f(x)$ 在 $[a,b]$ 上黎曼可积的充分必要条件是 $f(x)$ 在 $[a,b]$ 上几乎处处连续，即不连续的点的全体组成一个零测度集。

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\item  %Problem 03、S5.5定理2
设函数 $f(x)$ 在区间 $[a,b]$ 上是黎曼可积的。则 $f(x)$ 在 $[a,b]$ 上是勒贝格可积的，且
$$(L)\int _{[a,b]} f(x)dx = (R) \int_a^b f(x)dx. $$

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%\item  %Problem 04、S5.5定理3
%设 $f(x)$ 是 $[a,\infty)$ 上的一个非负实函数。
%设对任意的 $A>a$, 函数 $f(x)$ 在 $[a,A]$ 上是黎曼可积的，且黎曼反常积分 $$(R)\int_a^{\infty} f(x)dx$$ 收敛，
%则 $f(x)$ 在 $[a,\infty)$ 上是勒贝格可积的，且 
%$$(L)\int _{[a,\infty]} f(x)dx = (R) \int_a^\infty f(x)dx. $$
%
%\vspace{0.1cm}

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\item  %Problem 05、S5.5例子
设 $f(x) = \left\{
\begin{array}{ll}
\frac{\sin x}{x}, & x>0, \\ 
1, & x=0,
\end{array}
\right. $
则 $f(x)$ 在 $[0,\infty)$ 上连续，黎曼反常积分存在，但不是勒贝格可积的。

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%\item  %Problem 06、S5.6 勒贝格积分的几何意义、富比尼定理
%什么是 $\mathbb{R}^n$ 中的点集的直积 $A\times B$?  什么是 $\mathbb{R}^n$ 中的点集的截面？
%
%
%\vspace{0.1cm}

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\item  %Problem 07、S5.6定理1
（截面定理）设 $E\subseteq \mathbb{R}^{p+q}$ 是可测集。则对于 $\mathbb{R}^p$ 中的几乎所有的点 $x$, 截面 $E_x$ %$E_x=\{y\in\mathbb{R}^q:(x,y)\in E \}$ 
是 $\mathbb{R}^q$ 中的可测集，
而且测度 $m(E_x)$ 是 $x\in \mathbb{R}^p$ 上几乎处处有定义的可测函数，其积分就是 $E$ 的测度，即
$$m(E) = \int_{\mathbb{R}^p} m(E_x)dx. $$ 

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\item  %Problem 08、S5.6定理2
设 $A,B$ 分别是 $\mathbb{R}^p, \mathbb{R}^q$ 中的可测集，则 $A\times B$ 是 $\mathbb{R}^{p+q}$ 中的可测集，且 $$m(A\times B) = m(A)m(B).$$

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\item  %Problem 09、S5.6定理3
（非负可测函数的积分的几何意义）
设函数 $f(x)$ 是可测集 $E\subseteq \mathbb{R}^n$ 上的非负函数，
记 $$G(E,f) = \{(x,z): x\in E, 0\le z< f(x)\},$$ 
称为 $f(x)$ 在 $E$ 上的下方图形。则 $f(x)$ 是 $E$ 上的可测函数的充分必要条件是 $G(E,f)$ 是 $\mathbb{R}^{n+1}$ 中的可测集，此时 $f(x)$ 的积分就是下方图形的测度，即 $$\int_E f(x)dx = m(G(E,f)). $$

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\item  %Problem 10、S5.6定理4
（富比尼定理）
%\begin{enumerate}
%\item  
设函数 $f(P)=f(x,y)$ 在可测集 $A\times B\subseteq \mathbb{R}^{p+q}$ 上非负可测，
则对几乎处处 $x\in A$, $f(x,y)$ 作为 $y$ 的函数在 $B$ 上可测，且 
$$\int_{A\times B} f(P)dP = \int_Adx \int_B f(x,y)dy. $$ 

%\item  设 $f(P)=f(x,y)$ 在 $A\times B\subset \mathbb{R}^{p+q}$ 上可积，
%则对几乎处处 $x\in A$, $f(x,y)$ 作为 $y$ 的函数在 $B$ 上可积，
%$\int_B f(x,y)dy$ 作为 $x$ 的函数在 $A$ 上可积，且 
%$$\int_{A\times B} f(P)dP = \int_Adx \int_B f(x,y)dy. $$
%
%\end{enumerate}

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\item  %Problem 11、S5.6例子
证明函数 $f(x,y)=\frac{x^2-y^2}{(x^2+y^2)^2}$ 在 $E=(0,1)\times (0,1)$上不是勒贝格可积的。

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\item  %Problem 17、S5习题24
设 $\{r_k\}$ 是 $[0,1]$ 中的全体有理数，证明级数 $\sum\limits_{k=1}^{\infty} \frac{1}{k^2\sqrt{|x-r_k|}}$ 在 $[0,1]$ 上几乎处处收敛。

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\item  %Problem 18、S5习题25
设 $f(x)$ 是 $\mathbb{R}$ 上的勒贝格可积函数，证明级数 $\sum\limits_{n=1}^{\infty} f(x+n)$ 在 $\mathbb{R}$ 上几乎处处绝对收敛。

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\end{enumerate}


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\end{document}

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